Magic 3-D Polygons & Graphs

On this page
we will digress a little and consider 3 and 4 dimensional objects and other
types of graphs.
Actually, many of these are not polygons, but can have similar magical
properties to perimeter magic polygons.
I have taken many of thee drawings from [1], and present them with a minimum of
descriptive text.

Magic
The first eight digits can be placed on the corners (vertices) of a cube in 1680
different ways. It is possible to place them in such a manner that the sum of
the four integers on each of the six faces sum to the same value.
It turns out that the there is only one set of two groups of four that can be
used to construct an order-2 face magic cube. That is 1, 4, 6, 7 and 2, 3, 5, 8.
These two sets can be placed on opposite faces of the cube to produce 6
different order-2, face magic solutions. All have face sums of 18.
[2]Figure A.
is one of the six solutions.

Anti-magic
It is possible to place the digits 1 to 8 on the vertices of a cube in
such a manner that each face sums to a different value. If these sums are
in arithmetic progression, the cube is anti-magic.
Figures B and C are two of the 13 possible solutions. 12 of these consist
of 6 complement pairs, one is self-complementing.

There are no edge-magic or
anti-magic order-2 cubes. However, the sums of 11 of the 12 edges of Figure D
are prime numbers (but with duplication).
[3]

Magic
There are no order-2 (Trigg calls these V-type) edge magic tetrahedrons.
[3]

There are no
order-3 edge magic tetrahedrons (using consecutive integers)
Figures A, B, and C illustrate three non-normal solutions. A does not use the
integer 4, B. does not use the integers 1 or 11, C. Does not use the integer 2.

Anti-magic
There are no order-2 edge anti-magic tetrahedrons.
There are a total of 376 edge anti-magic tetrahedrons of order 3 (Trigg
calls these VM-type) with edge sums in arithmetic progression. Figure D.
is one solution .
Figure E is a richer solution (1 of 16 of this type). The edge sums are 7,
8, 9, 10, 11, 12, and the vertex only sums (i.e. order-2) are 3, 4, 5, 6;
giving ten consecutive sums!

Magic
If one digit is placed at each edge midpoint (none at the vertices), the
polygon is trivially edge anti-magic.
The first12 integers may be placed in pairs near the midpoints of the
edges (of the tetrahedron), so that all edges sum the same. Figure A

Anti-magic
There are a large number of edge anti-magic tetrahedrons of this type. B
and C show two of them.

As all the faces of a polyhedron are triangles, it is impossible to
place distinct integers on its vertices so that every triangle will have
the same perimeter sum. Therefore, no such polyhedron can be magic.
However, if the perimeter sums are all different the polyhedron will be
perimeter anti-magic.

In the case of an octahedron it turns out that there are 15 basic ways
to distribute the digits 1 to 6 on the vertices with eight different
perimeter sums. Of these 15, five produce distinct perimeter sums for the
eight triangles and so are perimeter anti-magic.

Figures A. and B. may each be complemented by subtracting each number
from 7, to provide another solution. C. is self-complementing.

D. shows the basic way to arrange the integers 1 to 4 on the vertices
of a tetrahedron. The perimeter sums are in the arithmetic progression
order 6, 7, 8, 9, making this a true, face anti-magic tetrahedron. The
edge sums of D are almost unique, meaning this is also an almost edge
anti-magic tetrahedron!

These figures are order-2 (2 integers per edge).Investigations into order-3 (and higher) figures of this type
are still required.

The corner numbers of these
16 parallelograms all sum to 34 (i.e. face magic), as do the 4
numbers in each of the four vertical lines. They were mapped from
the order 4 pantriagonal magic square shown.

Shown here is a Schlegel
diagram (graph) of the platonic solid 12 faced dodecahedron. The
numbers 1 to 25 are placed at the vertices of the faces in such a
way that all 12 faces are magic. The diagram is placed inside a
circle and 5 radial lines have been added. These lines and the
circle also sum to the constant.

Your aid I want, nine trees to plant
In rows just half a score;
And let there be in each row three.
Solve this: I ask no more.

In 1821, John Jackson published this ditty in
Rational Amusements for Winter Evenings.

Figure A. is the answer to Jackson’s little poem.

Tree-planting graphs are so-called because
they result from the effort to arrange as many “trees” as possible in a minimum
number of lines, each containing the same number of “trees”. This problem was
made popular by H. E, Dudeney (but introduced by John Jackson 75 years earlier).
See [8] for a good introduction to this
type of problem, although you will find it mentioned in many books on
recreational mathematics.

In keeping to the theme of magic
perimeters and graphs, I like to replace the ‘trees” with consecutive integers
in such a way that the lines are magic. Figure B. is the order 3 magic
square with 8 lines of 3 numbers all adding to the same constant.

The figure below illustrates an
order-5 magic star I have mapped to 5 other isomorphic “tree-planting” graphs.
[1]
These 6 graphs all have 10 numbers that appear in 5 lines of 4 numbers. Notice
however, that unlike most examples I have shown, these 10 numbers are not
consecutive. It is impossible to place the numbers 1 to 10 on a 5-pointed star
so that all lines sum the same.

A graph with q edges is said to be
super-magic if it is possible to label the edges with the numbers 1, 2, 3, …, q
in such a way that at each vertex v the sum of the labels on the edges incident
with v is the same.

Many super-magic graphs are isomorphic
to magic squares, as the following examples illustrate. Solid vertices in these
graphs represent the rows of the magic square., hollow vertices the columns.
These graphs is bipartite because no two like vertices are directly connected by
an edge.

Graph - Anti-magic

A graph with q edges is said to be
anti-magic if it is possible to label the edges with the numbers 1, 2, 3, …, q
in such a way that at each vertex v the sum of the labels on the edges incident
with v is different.

Many anti-magic graphs are isomorphic to
magic squares, as the following example illustrates. This graph is isomorphic to
the order-4 anti-magic square shown in Anti-magic squares. Solid vertices in
this graph represent the rows of the magic square., hollow vertices the columns.

Note that unlike anti-magic squares, it
is not required that anti-magic graphs have the sums form a consecutive series.
In fact, for normal anti-magic squares, at least 1 of the two diagonals must sum
to a value in the middle of the series. In the case of this graph, the sums form
a series from 30 to 38 but with 34 missing.

In
this triangle, the number at the apex is the difference between the
two numbers on the base of the triangle.
This is the only order-5 absolute difference triangle using the
consecutive integers from 1 to 15.
There are two order-2, four order-3, and four order-4 such
triangles. It is thought that there are no absolute difference
triangles higher then order-5 using consecutive integers.
[10]

Figure
A. shows nested triangles with a single digit at the apex of each
triangle. If you consider the digit at each end of a line (ie.
adjacent vertices) forming two 2 digit numbers, one of these two
numbers will always be divisible by seven (in these examples). This
example is called a cycle 1 because it oscillates between identical
triangles.

Figure B. is an example of a
cycle 4 Fractal Triangle because, starting from the center there are
four different triangles before the center triangle repeats as the
outer triangle. The author conjectures that no longer cycles are
possible.

Another magic object related to
magic squares, graphs, etc. is the magic circle. Benjamin Franklin, when
talking about his famous magic squares, mentioned a magic circle derived
from one. See a recreation of this on my Franklin Magic Squares page.
[12]

Here I show a set of magic circles, all
related to the same order-4 pandiagonal magic square. It is number 469 in the
sorted list of all 880 order-4 basic magic squares. All 48 pandiagonal magic
squares of this order contain all the patterns shown in this set of four graphs.
This is not necessarily true for other orders of pandiagonal magic
squares.

A., in the figure, is the magic
square I use for this example. Shown are the basic requirements for all magic
squares (ie all rows, columns, and the two main diagonals must sum to a
constant. To be classed as pandiagonal, two other patterns are required. These
appear in parts C. and D.

B. Contains 9 small circles,
each of which connect to 4 numbers that sum to the constant 34. Also, shown in
B. are 4 medium sized circles and one large circle, all of which also connect
to 4 numbers summing to 34.

C. contains 2 diagonal ovals
which show a pattern required to make the square pandiagonal. C. also shows 3
vertical and 3 horizontal ovals illustrating another magic pattern

D. shows 8 sets of four numbers
connected by distorted ovals. All of these sets also sum to 34. Also shown are
4 sets of four numbers connected by triangles. This is the other pattern that
is a requirement of pandiagonal magic squares.

This set of 4 graphs illustrate a total
of 9 patterns (2+3+2+2) that appear in an order-4 pandiagonal magic
square.

All combinations of 4 numbers sum to the constant
34.

However,
there is one additional pattern in the magic square that is not shown in
diagrams B, C or D. Can you find it?

See a pandiagonal magic square generator using a
magic torus on my Unusual
Squares page.

[12] An elaborate magic circle
appears on my Franklin
page. And a Prime circle on my
More primes page.

Some additional patterns

The six numbers on each of the four circles sum to 39.

Each spoke (including the hub) and each rim section sum to 38.

The middle number on each line is the difference between the two
outside numbers.

Graphs A and B are self-explanatory. In the “3-D” graph at C, the sum
of the 5 lines at each node is 65.

A and B are from
[9], C is from
[1] and [9]

10 nodesThe 10 nodes in this graph are each
connected to all the others with lines (edges) labelled with the
consecutive numbers from 1 to 45. The 9 lines connected to each node
sum to the same constant 207. Dominic Olivastro explains how to
construct this type of graph in [16],
pages 121-125 (using a 6 node graph as an example).

Priming the cubeThe corners of this cube are labeled with the consecutive
integers from 0 to 7 in such a way that two adjacent corners
(vertices) always sum to a prime number.

[13] See another
prime number graph containing numbers 1 to 22 on my
Prime Numbers page.
"Magic Hexagons" and "12 Magic Circles-6 Magic Squares" are on my
More Squares page.

In 1972, Solomon Golomb
proposed a new type of graph numbering called Graceful Graphs. [14]
In it, the nodes are numbered in such a way that the edges
connecting them form a set of consecutive numbers starting with 1.
The lowest node number is zero, and by necessity, the node numbers
are not consecutive. For a graft to be graceful, the highest node
number, the highest edge number, and the number of edges must all be
equal.

Figure A (below) is a
tetrahedron and figure C is a Schlegel representation of a cube.
Notice in figure B, one line seems to be missing. If that line is
added, the diagram becomes one of only 3 ungraceful graphs
with fewer then 6 nodes!

These graceful graphs are all from Martin Gardner's Mathematical
Games column in Scientific American Magazine .
[14] [15]

Tree Problems
These are my patterns. Can you assign consecutive
numbers to them so that each line equals the same value?
several more tree planting problems appear on my
Magic Star Updates
page.

Graceful graph problemsThese three graphs are taken from Gardner's column
[14] [15] and presented here as
problems. His column provides the solutions.
Figure A will use edge numbers 1 to 9 and B will use 1 to 22. C will use
numbers 1 to 25 for the edges and 0 to 25 for the nodes.